In 1920s, Courant's nodal domain theorem was proved by Richard Courant [1]}. The theorem states that the nodal lines of the \(k\) -th eigenfunction \(f_k\)  of a self-adjoint second order elliptic differential operator can not divide the domain \(D\)  into more than \(k\)  different subdomains. Here the nodal lines are refered to the set of zeros (nodes) of eigenfunctions and the subdomains are now known as the nodal domains. This is a basic result in spectral theory with wide applications. Courant's theorem can be considered as a natural generalization of Sturm's oscillation theorem for second order ODEs that the zeros of the \(k\) -th eigenfunction of a vibrating string divide the string into exactly \(k\)  subintervals. There are abundant extensions of Courant's theorem to non-linear operators like \(p\) -Laplacians, and to Riemannian manifolds and discrete settings, see, e.g., [2]}, [3]}, [4]}, [5]}.
